Analysis of Hessian Scaling for Local and Global Costs in Variational Quantum Algorithm

Barren plateaus in variational quantum algorithms are typically described by gradient concentration at random initialization. In contrast, rigorous results for the Hessian, even at the level of entry-wise variance, remain limited. In this work, we analyze the scaling of Hessian-entry variances at initialization. Using exact second-order parameter-shift identities, we write $H_{jk}$ as a constant-size linear combination of shifted cost evaluations, which reduces ${\rm Var}_\rho(H_{jk})$ to a finite-dimensional covariance--quadratic form. For global objectives, under an exponential concentration condition on the cost at initialization, ${\rm Var}_\rho(H_{jk})$ decays exponentially with the number of qubits $n$. For local averaged objectives in bounded-depth circuits, ${\rm Var}_\rho(H_{jk})$ admits polynomial bounds controlled by the growth of the backward lightcone on the interaction graph. As a consequence, the number of measurement shots required to estimate $H_{jk}$ to fixed accuracy inherits the same exponential (global) or polynomial (local) scaling. Extensive numerical experiments over system size, circuit depth, and interaction graphs validate the predicted variance scaling. Overall, the paper quantifies when Hessian entries can be resolved at initialization under finite sampling, providing a mathematically grounded basis for second-order information in variational optimization.